Abstract
Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by Berkolaiko, Kennedy, Kurasov, and Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by Kac and Pivovarchik in 2011, we characterize the graphs exhibiting maximally degenerate eigenvalues which are the figure-of-eight graph, the 3-watermelon graph, and the lasso trees—namely, trees decorated with lasso graphs.
Highlights
Introduction and notation we present the essentials about metric graphs used in the present text, for a more general introduction we refer to [4] and [5], see [6] and the recent [7].Metric graphs
We present the essentials about metric graphs used in the present text, for a more general introduction we refer to [4] and [5], see [6] and the recent [7]
Metric graphs are constructed as the quotient space of a set of distinct real intervals under an equivalence relation on the set of their endpoints
Summary
We present the essentials about metric graphs used in the present text, for a more general introduction we refer to [4] and [5], see [6] and the recent [7]. The quotient space G = E/ ∼ is a metric graph, whose set of edges and vertices are E = E(G) and, respectively, V = V (G) =. We use the generic term graph to refer to both metric and quantum graph whenever the set of Dirichlet vertices and the associated Laplacian are clear from the context. In the study of the spectral estimates of quantum graphs several techniques have been developed, many of which put in relation modifications of the metric graph with the changes occurring in the spectrum. Several of these are discussed in [9] under the name of surgery principles. In line with the previous paragraph, any modification brought to a graph G and its set of Dirichlet vertices D should be reflected in the associated Laplacian
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